The Euler method does not take into account the curvature of the solution, so it tends to give different results depending on the step size. RK, depending on the order, takes into account the curvature. And this makes the estimated "next step" more accurate. Basically, if you are pretending a straight line is a good approximation of a curve (Euler) you will always be overshooting your solution. But if you take account of the curvature (RK) you can follow the curve.
Compare with the famous hockey quote (Gretzky): Euler skates to where the puck is; Runge-Kutta skates to where the puck is going to be.
I recommend you read about these algorithms. You can start with Wikipedia articles here and here.
